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Primes of the form x² + 5y²

Shobhit

Member
Nov 12, 2013
23
If $p$ is an odd prime expressible as
$$p=x^2+5y^2$$
where $x,y$ are integers, then prove that $p \equiv 1 \text{ or }9\text{ (mod 20)}$.
 

mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
Re: Primes of the form $x^2+5y^2$

Very nice, I like this one. The proof follows immediately from basic genus theory, and I am not going to post it down here (spoiler purpose).

For beginners, I would give a hint :

A prime of the particular form is of the form 1, 3, 7 or 9 modulo 20


Balarka
.
 

chisigma

Well-known member
Feb 13, 2012
1,704
Re: Primes of the form $x^2+5y^2$

If $p$ is an odd prime expressible as
$$p=x^2+5y^2$$
where $x,y$ are integers, then prove that $p \equiv 1 \text{ or }9\text{ (mod 20)}$.
It must be necessarly x even and y odd or x odd and y even. In the first case You set x = 2 n and y = 2 m + 1 so that is...

$\displaystyle p = 4\ n^{2} + 20\ m^{2} + 20\ m + 5\ (1)$

... and then...

$\displaystyle p \equiv 4\ n^{2} + 5\ \text{mod}\ 20\ (2)$

Observing (2) it is evident that $\displaystyle p \equiv 1\ \text {mod}\ 20$ or $\displaystyle p \equiv 9\ \text {mod}\ 20$. The same result You obtaining supposing x odd and y even...


Kind regards

$\chi$ $\sigma$
 
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mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
Re: Primes of the form $x^2+5y^2$

How does (2) follows from (1)?
 

chisigma

Well-known member
Feb 13, 2012
1,704
Re: Primes of the form $x^2+5y^2$

How does (2) follows from (1)?
I made a mistake writing $\displaystyle p \equiv 4\ n^{2}\ \text{mod}\ 20$ instead of $\displaystyle p \equiv 4\ n^{2} + 5\ \text{mod}\ 20$, so that I corrected it... very sorry! (Angry) ...

Kind regards

$\chi$ $\sigma$
 

mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
Re: Primes of the form $x^2+5y^2$

Ah, I see, I have interpreted the question wrong. I interpreted is as : p is an odd prime expressible as x^2 + 5y^2 if and only if p is 1 or 9 modulo 20.

But perhaps OP must have intended exactly this? Can Shobhit clarify whether he did a typo or not?
 

Shobhit

Member
Nov 12, 2013
23
Re: Primes of the form $x^2+5y^2$

Ah, I see, I have interpreted the question wrong. I interpreted is as : p is an odd prime expressible as x^2 + 5y^2 if and only if p is 1 or 9 modulo 20.

But perhaps OP must have intended exactly this? Can Shobhit clarify whether he did a typo or not?
Balarka, the question is correct. You had interpreted it correctly.
 

mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
Re: Primes of the form $x^2+5y^2$

Shobhit said:
Balarka, the question is correct. You had interpreted it correctly.
I see, thank you for clarifying.

One can carry on with my hints on the #2 then.
 
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